Incorporating Level Set Methods in Geographical Information Systems (GIS) for Land-Surface Process Modeling

Transcription

1 Incorporaing Level Se Mehods in Geographical Informaion Sysems (GIS) for Land-Surface Process Modeling D. Pullar Geography Planning and Archiecure, The Universiy of Queensland, Brisbane QLD 4072, Ausralia Absrac: Land-surface processes include a broad class of models ha operae a a landscape scale. Curren modelling approaches end o be specialised owards one ype of process, ye i is he ineracion of processes ha is increasing seen as imporan o obain a more inegraed approach o land managemen. This paper presens a echnique and a ool ha may be applied generically o landscape processes. The echnique racks moving inerfaces across landscapes for processes such as waer flow, biochemical diffusion, and plan dispersal. Is heoreical developmen applies a Lagrangian approach o moion over a Eulerian grid space by racking quaniies across a landscape as an evolving fron. An algorihm for his echnique, called level se mehod, is implemened in a geographical informaion sysem (GIS). I fis wih a field daa model in GIS and is implemened as operaors in map algebra. The paper describes an implemenaion of he level se mehods in a map algebra programming language, called MapScrip, and gives example program scrips for applicaions in ecology and hydrology. Keywords: Spaial dynamics; Map algebra; GIS; Modelling 1. INTRODUCTION Over he pas decade here has been an explosion in he applicaion of models o solve environmenal issues. Many of hese models are specific o one physical process and ofen require exper knowledge o use. Increasingly generic modeling frameworks are being sough o provide analyical ools o examine and resolve complex environmenal and naural resource problems. These sysems consider a variey of land condiion characerisics, ineracions and driving physical processes. Variables accouned for include climae, opography, soils, geology, land cover, vegeaion and hydro-geography [Moore e al., 1993]. Physical ineracions include processes for climaology, hydrology, opographic landsurface/sub-surface fluxes and biological/ecological sysems [Sklar and Cosanza, 1991]. Providing a generic environmenal modeling framework for pracical environmenal issues is challenging. I does no exis now despie an overwhelming demand because here are deep echnical challenges o build inegraed modeling frameworks in a scienifically rigorous manner. I is his challenge his research addresses Background for Approach The paper describes a generic environmenal modeling language inegraed wih a Geographical Informaion Sysem (GIS) which suppors spaialemporal operaors o model physical ineracions occurring in wo ways. The rivial case where ineracions are isolaed o a locaion, and he more common and complex case where ineracions propagae spaially across landscape surfaces. The programming language has a srong heoreical and algorihmic basis. Theoreically, i assumes a Eulerian represenaion of sae space, bu propagaes quaniies across landscapes using Lagrangian equaions of moion. In physics, a Lagrangian view focuses on how a quaniy (waer volume or paricle) moves hrough space, whereas an Eulerian view focuses on a local fixed area of space and accouns for quaniies moving hrough i. The benefi of his approach is ha an Eulerian perspecive is eminenly suied o represening he variaion of environmenal phenomena across space, bu i is difficul o concepualise soluions for he equaions of moion and has compuaional drawbacks [Press e al, 1992]. On he oher hand, he Lagrangian view is ofen no favoured because i requires a global soluion ha makes i difficul

2 o accoun for local variaions, bu has he advanage of solving equaions of moion in an inuiive and numerically direc way. The research will address his dilemma by adoping a novel approach from he image processing discipline ha uses a Lagrangian approach over an Eulerian grid. The approach, called level se mehods, provides an efficien algorihm for modeling a naural advancing fron in a hos of seings [Sehian, 1999]. The reason he mehod works well over oher approaches is ha he advancing fron is described by equaions of moion (Lagrangian view), bu compuaionally he fron propagaes over a vecor field (Eulerian view). Hence, we have a very generic way o describe he moion of quaniies, bu can explicily solve heir advancing properies locally as propagaing zones. The research work will adap his echnique for modeling he moion of environmenal variables across ime and space. Specifically, i will add new daa models and operaors o a geographical informaion sysem (GIS) for environmenal modeling. This is considered o be a significan research imperaive in spaial informaion science and echnology [Goodchild, 2001]. The main focus of his paper is o evaluae if he level se mehod (Sehian, 1999) can: provide a heoreically and empirically supporable mehodology for modeling a range of inegral landscape processes, provide an algorihmic soluion ha is no sensiive o process iming, is compuaionally sable and efficien as compared o convenional explici soluions o diffusive processes models, be developed as par of a generic modelling language in GIS o express inegraed models for naural resource and environmenal problems? The ouline for he paper is as follow. The nex secion will describe he heory for spaialemporal processing using level ses. Secion 3 describes how his is implemened in a map algebra programming language. Two applicaion examples are given an ecological and a hydrological example o demonsrae he use of operaors for compuing reacive-diffusive ineracions in landscapes. Secion 4 summarises he conribuion of his research. 2. THEORY 2. 1 Inroducion Level se mehods [Sehian, 1999] have been applied in a large collecion of applicaions including, physics, chemisry, fluid dynamics, combusion, maerial science, fabricaion of microelecronics, and compuer vision. Level se mehods compue an advancing inerface using an Eulerian grid and he Lagrangian equaions of moion. They are similar o cos disance modeling used in GIS [Burroughs and McDonnell, 1998] in ha hey compue he spread of a variable across space, bu he moion is based upon parial differenial equaions relaed o he physical process. The advancemen of he inerface is compued hrough ime along a spaial gradien, and i may expand or conrac in is exen. See Figure 1. Figure 1. Shows a) a propagaing inerface parameerised by differenial equaions, b) inerface frons have variable inensiy and may expand or conrac based on field gradiens and driving process Theory The advanage of he level se mehod is ha i models moion along a sae-space gradien. Level se mehods sar wih he equaion of moion, i.e. an advancing fron wih velociy F is characerised by an arrival surface T(x,y). Noe ha F is a velociy field in a spaial sense. If F was consan his would resul in an expanding series of circular frons, bu for differen values in a velociy field he fron will have a more conored appearance as shown in Figure 1b. The moion of his inerface is always normal o he inerface boundary, and is progress is regulaed by several facors: F = ƒ(l, G, I) x y, = F n where L = local properies ha deermine he shape of advancing fron, G = global properies relaed o governing forces for is moion, I = independen properies ha regulae and influence he moion.

3 If he advancing fron is modeled sricly in erms of he movemen of eniy paricles, hen a sraighforward velociy equaion describes is moion: T F = 1 given T 0 = 0 where he arrival funcion T(x,y) is a ravel cos surface, and T 0 is he iniial posiion of he inerface. Insead we use level ses o describe he inerface as a complex funcion. The level se funcion φ is an evolving fron consisen wih he underlying viscosiy soluion defined by parial differenial equaions. This is expressed by he equaion: φ + F φ = 0 given φ(x,y,=0) where φ is a complex inerface funcion over ime period 0..n, i.e. φ(x,y, = 0.. n, φ is he spaial and emporal derivaives for viscosiy equaions. The Eulerian view over a spaial domain imposes a discreisaion of space, i.e. he raser grid, which records changes in value z. Hence, he level se funcion becomes φ(x,y,z,) o describe an evolving surface over ime. Furher deails are given in Sehian [1999] along wih efficien algorihms. The nex secion describes he inegraion of he level se mehods wih GIS. 3. MAP ALGEBRA MODELLING compuaions has a special significance. For insance, processes ha involve spreading or ranspor acing along environmenal gradiens wihin he landscape. Therefore special conrol needs o be exercised on he order of execuion. Burrough [1998] describes wo exra conrol mechanisms for diffusion and direced opology. Figure 2 shows he hree principle ypes of processing orders, and hey are: row scan order governed by he clockwork laice srucure, spread order governed by he spreading or scaering of a maerial from a more concenraed region, flow order governed by advecion which is he ranspor of a maerial due o velociy. a) Row scan order b) Diffuse (spread) order 3. 1 Map Algebra Spaial models are wrien in a map algebra programming language. Map algebra is a funcionoriened language ha operaes on four implici spaial daa ypes: poin, neighbourhood, zonal and whole landscape surfaces. Surfaces are ypically represened as a discree raser where a poin is a cell, a neighbourhood is a kernel cenred on a cell, and zones are groups of cells. Common examples of raser daa include errain models, caegorical land cover maps, and scalar emperaure surfaces. Map algebra is used o program many ypes of landscape models ranging from land suiabiliy models o mineral exploraion in he geosciences [Burrough and McDonnell, 1998] [Bonham-Carer, 1994]. The synax for map algebra follows a mahemaical syle wih saemens expressed as equaions. These equaions use operaors o manipulae spaial daa ypes for poin and neighbourhoods. Expressions ha manipulae a raser surface may use a global operaion or alernaively ierae over he cells in a raser. For insance he GRID map algebra [Gao e al., 1993] defines an ieraion consruc, called docell, o apply equaions on a cell-by-cell basis. This is rivially performed on columns and rows in a clockwork manner. However, for environmenal phenomena here are siuaions where he order of c) Advecive (flow) order Figure 2. Spaial processing orders for raser Our implemenaion of map algebra, called MapScrip [Pullar, 2001], includes a special ieraion consruc ha suppors hese processing orders. MapScrip is a lighweigh language for processing raser-based GIS daa using map algebra. The language parser and engine are buil as a sofware componen o ineroperae wih he IDRISI GIS [Easman, 1997]. MapScrip is buil in C++ wih a class hierarchy based upon a value ype. Varians for value ypes include numerical, boolean, emplae, cells, or a grid. MapScrip suppors combinaions of hese daa ypes wihin equaions wih basic arihmeic and relaional comparison operaors. Algebra operaions on emplaes ypically resul in an aggregae value assigned o a cell [Pullar, 2001]; his is similar o he convoluion inegral in image algebras [Rier

4 e al., 1990]. The language suppors ieraion o execue a block of saemens in hree ways: a) docell consruc o process raser in a row scan order, b) dospread consruc o process raser in a spread order, c) doflow o process raser by flow order. Examples are given in subsequen secions. Process models will also involve a iming loop which may be handled as a general while(<condiion>)..end consruc in MapScrip where he condiion expression includes a sysem ime variable. This ime variable is used in a specific fashion along wih a sysem ime sep by cerain operaors, namely diffuse() and fluxflow() described in he nex secion, o model diffusion and advecion as a ime evolving fron. The evolving fron represens quaniies such as vegeaion growh or surface runoff. 3.2 Ecological Example This secion presens an ecological example based upon plan dispersal in a landscape. The populaion of a species follows a conrolled growh rae and a he same ime spreads across landscapes. The heory of he rae of spread of an organism is given in Tilman and Kareiva [1997]. The area occupied by a species grows log-linear wih ime. This may be modelled by coupling a spaial diffusion erm wih an exponenial populaion growh erm; he combinaion produces he familiar reaciondiffusion model. A simple growh populaion model is used where he reacion erm considers one populaion conrolled by birhs and moraliies is: dn d N = r (1 ) (1) K where N is he size of he populaion, r is he rae of change of populaion given in erms of he difference beween birh and moraliy raes, and K is he carrying capaciy. Furher discussion of populaion models can be found in Jørgensen and Bendoricchio [2001]. The diffusive erm spreads a quaniy hrough space a a specified rae: du d du = dx D (2) dx where u is he quaniy which in our case is populaion size, and D is he diffusive coefficien. The model is operaed as a coupled compuaion. Over a discreized space, or raser, he diffusive erm is esimaed using a numerical scheme [Press e al., 1992]. The disance over which diffusion akes place in ime sep d is minimally consrained by he raser resoluion. For a sable compuaional process he following condiion mus be saisfied: d 2D 1 (3) 2 dx This basically saes ha o accoun for he diffusive process, he erm 2D dx is less han he velociy of he advancing fron. This would no be difficul o compue if D is consan, bu is problemaic if D is variable wih respec o landscape condiions. This problem may be overcome by progressing along a diffusive fron over he discree raser based upon disance raher han being consrained by he cell resoluion. The processing and diffusive operaor is implemened in a map algebra programming language. The code fragmen in Figure 3 shows a map algebra scrip for a single ime sep for he coupled reacive-diffusion model for populaion growh. while (ime < 100) dospread pop = pop + (diffuse(kernel*pop)) pop = pop + (pop * (1 - (pop/k)) enddo end where he diffusive consan is sored in he kernel: kernel = Figure 3. Map algebra scrip and convoluion kernel for populaion dispersion. The variable pop is a raser, K and D are consans, and he kernel is a 3x3 emplae. I is assumed a ime sep is defined and he scrip is run in a simulaion. The firs line conained in he nesed cell processing consruc (i.e. dospread) is he diffusive erm and he second line is he populaion growh erm. The operaor of ineres in he scrip shown in Figure 3 is he diffuse operaor. I is assumed ha he scrip is run wih a given ime sep. The operaor uses a sysem ime sep which is compued o balance he effec of process errors wih efficien compuaion. Wih knowledge of he ime sep he ieraive consruc applies an appropriae disance propagaion such ha he condiion in Equaion 3 is no violaed. The level se algorihm [Sehian, 1999] is used o do his in a sable and accurae way. As a diffusive fron propagaes hrough he raser, a cos disance kernel assigns he proper ime o each raser cell. The ime assigned o he cell corresponds o he minimal cos i akes o reach ha cell. Hence cell processing is conrolled by propagaing he kernel ouward a a speed adapive o he local conex raher han meeing an arbirary global consrain.

5 3.3 Hydrological Example This secion presens a hydrological example based upon surface dispersal of excess rainfall across he errain. The movemen of waer is described by he coninuiy equaion: h = e q (4) where h is he waer deph (m), e is he rainfall excess (m/s), q is he discharge (m/hr) a ime. Discharge is assumed o have seady uniform flow condiions, and is deermined by Manning s equaion: q 1 h n s = v h = (5) where q is he flow velociy (m/s), h is waer deph, and s is he surface slope (m/m). An explici mehod of calculaion is used o compue velociy and deph over raser cells, and equaions are solved a each ime sep. A conservaive form of a finie difference mehod solves for q in Equaion 5. To simplify discussions we describe quasi-onedimensional equaions for he flow problem. The acual numerical compuaions are normally performed on an Eulerian grid [Julien e al., 1995]. + curren cell (unknown) processed cells x x+ x Figure 4. Compuaion of curren cell (x+ x,,+ ). Finie-elemen approximaions are made o solve he above parial differenial equaions for he onedimensional case of flow along a srip of uni widh. This leads o a coupled model wih one erm o mainain he coninuiy of flow and anoher erm o compue he flow. In addiion, all calculaions mus progress from an uphill cell o he down slope cell. This is implemened in map algebra by a ieraion consruc, called doflow, which processes a raser by flow order. Flow disance is measured in cell size x per uni lengh. One srip is processed during a ime inerval (Figure 4). The conservaive soluion for he coninuiy erm using a firs order approximaion for Equaion 5 is derived as: qx+ x, qx, hx+ x + = hx+ x (6),, x where he inflow q x, and ouflow q x+ x, are calculaed in he second erm using Equaion 6 as: q x, = vx, h (7) The calculaions approximae discharge from previous ime seps. Discharge is dynamically deermined wihin he coninuiy equaion by waer deph. The rae of change in sae variables for Equaion 6 needs o saisfy a sabiliy condiion where v / x 1 o mainain numerical sabiliy. The physical inerpreaion of his is ha a finie volume of waer would flow across and ou of a cell wihin he ime sep. Typically he cell resoluion is fixed for he raser, and adjusing he ime sep requires resaring he simulaion cycle. Flow velociies change dramaically over he course of a sorm even, and i is problemaic o se an appropriae ime sep which is efficien and yields a sable resul. The hydrological model has been implemened in a map algebra programming language Pullar [2003]. To overcome he problem menioned above we have added high level operaors o compue he flow as an advancing fron over a landscape. The ime sep advances his fron adapively across he landscape based upon he flow velociy. The level se algorihm [Sehian, 1999] is used o do his in a sable and accurae way. The map algebra scrip is given in Figure 5. The imporan operaor is he fluxflow operaor. I compues he advancing fron for waer flow across a DEM by hydrological principles, and compues he local drainage flux rae for each cell. The flux rae is used o compue he ne change in a cell in erms of flow deph over an adapive ime sep. while (ime < 120) doflow(dem) fvel = 1/n * pow(deph,m) * sqr(grade) deph = deph + (deph * fluxflow(fvel)) enddo end Figure 5. Map algebra scrip for excess rainfall flow compued over a 120 minue even. The variables deph and grade are rasers, fvel is he flow velociy, n and m are consans in Manning s equaion. I is assumed a ime sep is defined and he scrip is run in a simulaion. The firs line in he nesed cell processing (i.e. doflow) compues he flow velociy and he second line compues he change in deph from he previous value plus any ne change (inflow ouflow) due o velociy flux across he cell.

6 4. CONCLUSION The level se mehod provides he following benefis: i more direcly models moion of spaial phenomena and may handle boh expanding and conracing inerfaces, is based upon differenial equaions relaed o he spaial dynamics of physical processes. Despie he poenial for using level se mehods in GIS and land-surface process modeling, here are no commercial or research sysems ha use his approach. Commercial sysems such as GRID [Gao e al., 1993], and research sysems such as PCRaser [Wesseling e al., 1996] offer flexible and powerful map algebra programming languages. Bu operaions ha involve reacion-diffusive processing are specific o one conex, such as groundwaer flow. We believe he level se mehod offers a more generic approach ha allows a user o program flow and diffusive landscape processes for a variey of applicaion conexs. We have shown ha i provides an appropriae heoreical underpinning and may be efficienly implemened in a GIS. We have demonsraed is applicaion for wo landscape processes albei relaively simple examples bu hese may be exended o deal wih more complex and dynamic circumsances. The validaion for improved environmenal modeling ools ulimaely ress in heir upake and usage by scieniss and engineers. The ool may be accessed from he web sie (version wih enhancemens available April 2004) for use wih IDRSIS GIS [Easman, 1997] and in he fuure wih ArcGIS. I is hoped ha a larger communiy of users will make use of he mehodology and implemenaion for a variey of environmenal modeling applicaions. 5. REFERENCES Bonham-Carer, G.F., Geographic Informaion Sysems for Geoscieniss: Modelling wih GIS Pergamon Elsevier Science, New York, Burrough, P.A., Dynamic Modelling and Geocompuaion. In: P.A. Longley e al., Geocompuaion: A Primer, Wiley, England, , Burrough, P.A., and R. McDonnell, Principles of Geographic Informaion Sysems, Oxford Universiy Press, New York, Easman, J.R., IDRISI for Windows Version 2.0, Clark Universiy, Worceser, Gao, P., C. Zhan, and S. Menon, An Overview of Cell-Based Modeling wih GIS. In M.F. Goodchild e al. (eds), Environmenal Modeling wih GIS, Oxford Universiy Press, , Goodchild, M., A Geographer Looks a Spaial Informaion Theory. in: Goos G., Hermanis J. and van Leeuwen J. (eds), COSIT Spaial Informaion Theory, LNCS 2205, 1-13, Jørgensen, S. and G. Bendoricchio, Fundamenals of Ecological Modelling. Elsevier, New York, Julien, P.Y., B. Saghafian, and F. Ogden, Raser- Based Hydrologic Modelling of Spaially- Varied Surface Runoff. Waer Resources Bullein, 31(3), , Moore, I.D., A. Turner, J. Wilson, S. Jenson and L. Band, GIS and Land-Surface-Subsurface Process Modeling. in: M.F.Goodchild e al. (eds), Environmenal Modeling wih GIS, Oxford Universiy Press, New York, Press, W., B. Flannery, S. Teukolsky, and W. Veerling, Numerical Recipes in C: The Ar of Scienific Compuing, 2 nd Ed. Cambridge Universiy Press, Cambridge, Pullar, D., MapScrip: A Map Algebra Programming Language Incorporaing Neighborhood Analysis. GeoInformaica 5(2), , Pullar, D., Simulaion Modelling Applied To Runoff Modelling Using MapScrip. Transacions in GIS, 7(2): , Rier, G., J. Wilson, and J. Davidson. Image Algebra: An Overview. Compuer Vision, Graphics, and Image Processing, 4, , Sehian, J.A., Level Se Mehods and Fas Marching Mehods. Cambridge Universiy Press, Cambridge, Sklar, F.H. and R. Cosanza, The Developmen of Dynamic Spaial Models for Landscape Ecology: A Review and Progress. in: M.G. Turner and R. Gardner (eds), Quaniaive Mehods in Ecology, Springer-Verlag, New York, , Tilman, D., and P. Kareiva, Spaial Ecology: The Role of Space in Populaion Dynamics and Inerspecific Ineracions. Princeon Universiy Press, Princeon, New Jersey, USA, Wesseling C.G., D. Karssenberg P.A. Burrough and W.P. van Deursen, Inegraing Dynamic Environmenal Models in GIS: The Developmen of a Dynamic Modelling Language. Transacions in GIS, 1(1), 40-48, 1996.